Toppling on inclined plane

2
A uniform solid cone has height 30 cm and base radius \(r \mathrm {~cm}\). The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted and the cone remains in equilibrium until the angle of inclination of the plane reaches \(35 ^ { \circ }\), when the cone topples. The diagram shows a cross-section of the cone.

1. Find the value of \(r\).
2. Show that the coefficient of friction between the cone and the plane is greater than 0.7 .

7 \includegraphics[max width=\textwidth, alt={}, center]{81411376-b926-4857-bc9b-ac85d7957f3d-3_327_1006_1037_573} A light container has a vertical cross-section in the form of a trapezium. The container rests on a horizontal surface. Grain is poured into the container to a depth of \(y \mathrm {~m}\). As shown in the diagram, the cross-section \(A B C D\) of the grain is such that \(A B = 0.4 \mathrm {~m}\) and \(D C = ( 0.4 + 2 y ) \mathrm { m }\).

1. When \(y = 0.3\), find the vertical height of the centre of mass of the grain above the base of the container.
2. Find the value of \(y\) for which the container is about to topple.

8 A rectangular lamina of mass \(M\) has vertices at the origin \(O ( 0,0 ) , A ( 24 a , 0 ) , B ( 24 a , 6 a )\) and \(C ( 0,6 a )\), where \(a\) is a positive constant. A small object \(P\) of mass \(m\) is attached to the lamina at the point ( \(x , y\) ). The centre of mass of the system consisting of the lamina and \(P\) is at the point ( \(\mathrm { x } , \mathrm { y }\) ). \(P\) is modelled as a particle and the lamina is modelled as being uniform.

1. Show that \(x = \frac { 12 M a + m x } { M + m }\).
2. Find a corresponding expression for \(\bar { y }\). The lamina, with \(P\) no longer attached, is placed on a horizontal rectangular table, with its sides parallel to the edges of the table, and partly overhanging the edges of the table, as shown in the diagram. The corner of the table is at the point ( \(6 a , 2 a\) ). \includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-6_538_1431_849_246} When \(P\) is placed on the lamina at \(O\), the lamina topples over one of the edges of the table.
3. Show that \(\mathrm { m } > \frac { 1 } { 2 } \mathrm { M }\). The lamina is now put back on the table in the same position as before. \(P\) is placed at the point \(( 12 a , 6 a )\) on the smooth upper surface of the lamina, and is projected towards \(O\). At a subsequent instant during the motion, \(P\) is at the point (12ak, 6ak) where \(0 < k < 1\).
4. Assuming that the lamina has not yet toppled, find, in terms of \(M\) and \(m\), the value of \(k\) for which the centre of mass of the system lies on the table edge parallel to \(O C\).
5. For the case \(\mathrm { m } = \frac { 3 } { 2 } \mathrm { M }\), determine which table edge the lamina topples over.

7 \includegraphics[max width=\textwidth, alt={}, center]{d1eb99a1-04e5-43bc-87b4-d0f7c962135c-4_353_579_248_744} The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a trapezium \(A B C D\) with \(A B\) and \(C D\) perpendicular to \(A D\). The lengths of \(A B\) and \(A D\) are each 5 cm and the length of \(C D\) is \(( a + 5 ) \mathrm { cm }\).

1. Show the distance of the centre of mass of the prism from \(A D\) is $$\frac { a ^ { 2 } + 15 a + 75 } { 3 ( a + 10 ) } \mathrm { cm } .$$ The prism is placed with the face containing \(A B\) in contact with a horizontal surface.
2. Find the greatest value of \(a\) for which the prism does not topple. The prism is now placed on an inclined plane which makes an angle \(\theta ^ { \circ }\) with the horizontal. \(A B\) lies along a line of greatest slope with \(B\) higher than \(A\).
3. Using the value for \(a\) found in part (ii), and assuming the prism does not slip down the plane, find the greatest value of \(\theta\) for which the prism does not topple.

3. \begin{figure}[h] \end{figure} Figure 1 shows a uniform triangular lamina \(A B C\) placed with edge \(B C\) along the line of greatest slope of a plane inclined at an angle \(\theta\) to the horizontal. The lengths \(A C\) and \(B C\) are 15 cm and 9 cm respectively and \(\angle A B C\) is a right angle.

1. Find the distance of the centre of mass of the lamina from
   1. \(\quad A B\),
   2. \(B C\). Assuming that the plane is rough enough to prevent the lamina from slipping,
2. find in degrees, correct to 1 decimal place, the maximum value of \(\theta\) for which the lamina remains in equilibrium.
   (4 marks)

6 Fig. 6.1 shows a solid uniform prism OABCDEFG . The \(\mathrm { Ox } , \mathrm { Oy }\) and Oz axes are also shown. The cross-section of the prism is a trapezium. Fig. 6.2 shows the face OABC . The dimensions shown in the figures are in centimetres. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The centre of mass of the prism has coordinates \(( \bar { x } , \bar { y } , \bar { z } )\).

1. Determine the values of \(\bar { x } , \bar { y }\) and \(\bar { z }\).
2. By considering triangle PBA, where P has coordinates ( \(\bar { x } , 0 , \bar { z }\) ), determine whether it is possible for the prism to rest with the face ABEF in contact with a horizontal plane without toppling.

1 A uniform solid rectangular prism has cross-section with width \(w \mathrm {~cm}\) and height 24 cm . Another uniform solid prism has cross-section in the shape of an isosceles triangle with width \(w \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The prisms are both placed with their axes vertical on a rough horizontal plane (see Fig. 1.1, which shows the cross-sections through the centres of mass of both solids). \begin{figure}[h] \end{figure} The plane is slowly tilted and both solids remain in equilibrium until the angle of inclination of the plane reaches \(\alpha\), when both solids topple simultaneously.

1. Determine the value of \(h\). It is given that \(w = 12\).
2. Determine the value of \(\alpha\). Both prisms are made from the same material and are of uniform density. The triangular prism is now placed on top of the rectangular prism to form a composite body C such that the base of the triangular prism coincides with the top of the rectangular prism. A cross-section of C is shown in Fig. 1.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6418c1b7-092a-4747-bc88-1b57815c6ad9-2_777_439_1784_258} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure}
3. Determine the height of the centre of mass of C from its base.

5 Fig. 5.1 shows a solid L-shaped ornament, of uniform density. The ornament is 3 cm thick. The \(x , y\) and \(z\) axes are shown, along with the dimensions of the ornament. The measurements are in centimetres. \begin{figure}[h] \end{figure}

1. Determine, with reference to the axes shown, the coordinates of the ornament's centre of mass. Fig. 5.2 shows the ornament placed so that the shaded face (indicated in Fig. 5.1) is in contact with a plane inclined at \(\theta ^ { \circ }\) to the horizontal, with the 4 cm edge parallel to a line of greatest slope. The surface of the plane is sufficiently rough so that the ornament will not slip down the plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-6_646_844_1452_242} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}
2. Determine the minimum and maximum possible values of \(\theta\) for which the ornament does not topple. The ornament is now placed with its shaded face in contact with a rough horizontal surface. A force of magnitude \(P\) N, acting parallel to the planes of the L -shaped faces, is applied to one of the edges of the ornament, as shown in Fig. 5.3. The force is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the ornament and the surface is \(\mu\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-7_524_680_452_246} \captionsetup{labelformat=empty} \caption{Fig. 5.3}
   \end{figure} The value of \(P\) is gradually increased until the ornament is on the point of toppling but does not slide.
3. Determine the minimum value of \(\mu\).
4. Explain how your answer to part (c) would change if the angle between \(P\) and the horizontal was less than \(30 ^ { \circ }\).

6 In this question you must show detailed reasoning. As shown in Fig. 6.1, the region R is bounded by the lines \(x = 1 , x = 2 , y = 0\) and the curve \(y = 2 x ^ { 2 }\) for \(1 \leq x \leq 2\). A uniform solid of revolution, S , is formed when R is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. \begin{figure}[h] \end{figure}

1. Show that the volume of S is \(\frac { 124 \pi } { 5 }\).
2. Show that the distance of the centre of mass of S from the centre of its smaller circular plane surface is \(\frac { 43 } { 62 }\). Fig. 6.2 shows S placed so that its smaller circular plane surface is in contact with a slope inclined at \(\alpha ^ { \circ }\) to the horizontal. S does not slip but is on the point of tipping. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-6_458_565_2014_694} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure}
3. Find the value of \(\alpha\), giving your answer in degrees correct to 3 significant figures.

7 Particles \(A , B\) and \(C\) of masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and 5 kg respectively are joined by light rigid rods to form a triangular frame. The frame is placed at rest on a horizontal plane with \(A\) at the point ( 0,0 ), \(B\) at the point ( \(0.6,0\) ) and \(C\) at the point ( \(0.4,0.2\) ), where distances in the coordinate system are measured in metres (see Fig. 1). \begin{figure}[h] \end{figure} \(G\), which is the centre of mass of the frame, is at the point \(( \bar { x } , \bar { y } )\).

1. - Show that \(\bar { x } = 0.38\).
   A rough plane, \(\Pi\), is inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 3 } { 5 }\). The frame is placed on \(\Pi\) with \(A B\) vertical and \(B\) in contact with \(\Pi\). \(C\) is in the same vertical plane as \(A B\) and a line of greatest slope of \(\Pi . C\) is on the down-slope side of \(A B\). The frame is kept in equilibrium by a horizontal light elastic string whose natural length is \(l \mathrm {~m}\) and whose modulus of elasticity is \(g \mathrm {~N}\). The string is attached to \(A\) at one end and to a fixed point on \(\Pi\) at the other end (see Fig. 2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{493f11f4-e25c-4eeb-a0ab-20ec6d7a7a7d-5_611_842_1649_248} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The coefficient of friction between \(B\) and \(\Pi\) is \(\mu\).
2. Show that \(l = 0.3\).
3. Show that \(\mu \geqslant \frac { 14 } { 27 }\). \section*{OCR} Oxford Cambridge and RSA

\includegraphics{figure_2} Figure 2 shows the region \(R\) bounded by the curve with equation \(y^2 = rx\), where \(r\) is a positive constant, the \(x\)-axis and the line \(x = r\). A uniform solid of revolution \(S\) is formed by rotating \(R\) through one complete revolution about the \(x\)-axis.

1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac{4}{5}r\). [6]

The solid is placed with its plane face on a plane which is inclined at an angle \(\alpha\) to the horizontal. The plane is sufficiently rough to prevent \(S\) from sliding. Given that \(S\) does not topple,

1. find, to the nearest degree, the maximum value of \(\alpha\). [4]

\includegraphics{figure_2} A uniform lamina occupies the region \(R\) bounded by the \(x\)-axis and the curve $$y = \sin x, \quad 0 \leq x \leq \pi,$$ as shown in Figure 2.

1. Show, by integration, that the \(y\)-coordinate of the centre of mass of the lamina is \(\frac{\pi}{8}\). [6]

\includegraphics{figure_3} A uniform prism \(S\) has cross-section \(R\). The prism is placed with its rectangular face on a table which is inclined at an angle \(\theta^{\circ}\) to the horizontal. The cross-section \(R\) lies in a vertical plane as shown in Figure 3. The table is sufficiently rough to prevent \(S\) sliding. Given that \(S\) does not topple,

1. find the largest possible value of \(\theta\). [3]

\(PQR\) is a triangular lamina with \(PQ = 18\) cm, \(QR = 24\) cm and \(PR = 30\) cm.

1. Verify that angle \(PQR\) is a right angle and find the distances of the centre of mass of the lamina from
   1. \(PQ\),
   2. \(QR\).
   [5 marks]

\includegraphics{figure_6} The lamina is held in a vertical plane and placed on a line of greatest slope of a rough plane inclined at an angle \(\theta\) to the horizontal, as shown.

1. Find the largest value of \(\theta\) for which equilibrium will not be broken by toppling. [4 marks]